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From a computing point of view, there is no difference between the power supply voltage for any computing system: 5V, 3.3V, 2.2V, 1.8V, 0.8V. They all compare, shift, add, etc etc, just the same.
But, from the silicon chip point of view, a lower voltage allows the use of smaller transistors and thus faster operation. A lower voltage also means that the logic levels have less of a voltage swing so they are faster (this is not always the case but in general it is).
The ultimate barrier to processor density, and hence processing power, is getting rid of heat from the chip, so by using a lower voltage you dissipate less power and can thus have a more dense and complex processor.
A lower processor power consumption means that the user power supply is lower power.
The downside of a low voltage processor is that, by definition, its output voltage signal is less so you may have to add circuits to drive 5V logic parts for example.
The other downside of a lower voltage is that the noise immunity is decreased.
A good example of a 5V processor unit is the Arduino Uno and a good example of a 3.3V processor unit is the Arduino Due. Although the Due is vastly superior to the Uno in all respects, the Due does not interface directly to most Arduino shields (input output modules or peripherals), which mainly run off 5V.
In terms of the future, it is almost certain that hobby processor modules and peripherals will move from 5V to 3.3V, I think anyway.